The mixing of $K^0-\bar{K^0}$, $D^0-\bar{D^0}$ and
$B_{(s)}^0-\bar{B^0_{(s)}}$ provides a sensitive probe to explore new physics
beyond the Standard Model. The scale invariant unparticle physics recently
proposed by Georgi can induce flavor-changing neutral current and contribute to
the mixing at tree level. We investigate the unparticle effects on
$B^0-\bar{B^0}$ and $D^0-\bar{D^0}$ mixing. Especially, the newly observed
$D^0-\bar{D^0}$ mixing sets the most stringent constraints on the coupling of
the unparticle to quarks.Comment: 9 pages, some errors corrected, published versio

Li, Xue-Qian

Wei, Zheng-Tao

English

arXiv

AbstractThe mixing of K0–K0¯, D0–D0¯ and B(s)0–B(s)0¯ provides a sensitive probe to explore new physics beyond the Standard Model. The scale invariant unparticle physics recently proposed by Georgi can induce flavor-changing neutral current and contribute to the mixing at tree level. We investigate the unparticle effects on B0–B0¯ and D0–D0¯ mixing. Especially, the newly observed D0–D0¯ mixing sets the most stringent constraints on the coupling of the unparticle to quarks

Elsevier - Publisher Connector 

Physics Letters B 651 (2007) 380–383
www.elsevier.com/locate/physletb
Unparticle physics effects on D0–D0 mixing
Xue-Qian Li, Zheng-Tao Wei ∗
Department of Physics, Nankai University, Tianjin 300071, China
Received 16 May 2007; received in revised form 4 June 2007; accepted 21 June 2007
Available online 23 June 2007
Editor: T. Yanagida
Abstract
The mixing of K0–K0, D0–D0 and B0
(s)
–B0
(s)
provides a sensitive probe to explore new physics beyond the Standard Model. The scale
invariant unparticle physics recently proposed by Georgi can induce flavor-changing neutral current and contribute to the mixing at tree level. We
investigate the unparticle effects on B0–B0 and D0–D0 mixing. Especially, the newly observed D0–D0 mixing sets the most stringent constraints
on the coupling of the unparticle to quarks.
© 2007 Elsevier B.V. Open access under CC BY license.1. Introduction
It is well known that scale invariance is broken by renormal-
ization and dimensional parameters in quantum field theories.
The concept of scale invariance (and more generally the confor-
mal symmetry) may still play an important role in high energy
physics. For an asymptotically free theory, such as QCD, the
scale invariance is recovered in the high energy limit. In the
concerned practical physics processes of high energies, break-
ing of scale invariance can be systematically incorporated in
the anomalous dimensions of operators using the renormaliza-
tion group method [1]. It is indicated that the scale invariance
in the infrared region may be quite different and less known [2].
But the idea of scale invariance is so simple and attractive that
there is no a priori to repel it from our world.
In [3], Georgi proposed that a scale invariant stuff contains
no particle, but the so-called unparticle. The unparticle pos-
sesses some properties which are different from that of ordinary
particles. The first aspect is that it has a non-trivial scale dimen-
sion dU . The dimension of unparticle is in general fractional
rather than an integral number (the dimension for a fermion is
half-integral). The fractional dimension must come from some
complicated dynamics whose details are unknown at present.
* Corresponding author.
E-mail addresses: lixq@nankai.edu.cn (X.-Q. Li), weizt@nankai.edu.cn
(Z.-T. Wei).0370-2693 © 2007 Elsevier B.V.
doi:10.1016/j.physletb.2007.06.045
Open access under CC BY license.Another aspect is that the free unparticle has no definite mass.
That means that the Lorentz-invariant four-momentum square
P 2 is not fixed for a real unparticle. Georgi observed that un-
particle with scale dimension looks like a non-integral number
dU of invisible massless particles [3]. To be consistent with the
present experimental observations, the coupling of unparticle
to the ordinary Standard Model (SM) matter must be suffi-
ciently weak. However, it may be relevant to the TeV physics
and might be explored at the LHC and ILC. The interactions be-
tween the unparticle and the SM particles are described in the
framework of low energy effective theory and lead to various in-
teresting phenomena. There have been some phenomenological
explorations on possible observable effects caused by unparti-
cles [3–10].
The mixing of K0–K0, D0–D0 and B0(s)–B
0
(s) is of fun-
damental importance to test the SM and explore new physics
beyond the SM. In the scenarios of new physics, there may ex-
ist a flavor-changing neutral current (FCNC) to result in such
a mixing which can only be realized via loops in the frame-
work of the SM. Thus this observable could be sensitive to
new physics effects. In fact, many authors used to explore ev-
idence of new physics in B0–B0 (or B0s –B0s ) mixing because
data about the mixing have been available for a long while. In
the proposed scenario [3], the unparticle can couple to different
flavors of quarks and induce FCNC even at tree level as long as
the unparticle is neutral. Thus it will cause new contributions to
the particle–antiparticle mixing, B0–B0, D0–D0 mixing. Gen-
X.-Q. Li, Z.-T. Wei / Physics Letters B 651 (2007) 380–383 381erally, based on physics conjecture, the energy scale concerning
unparticle is high that it should cause smaller influence on the
K0–K0 mixing, especially the SM contribution to the mix-
ing obviously dominates. The unparticle effects on B0(s)–B
0
(s)
mixing had been studied in [6,7] roughly. Since the B0(s)–B0(s)
mixing parameter xBd(s) is large and generally the contributions
from the SM dominate, and the new physics effect if it exists, is
less important, thus the observable is not so sensitive to the new
physics. Whereas for the D system, the SM contribution is con-
firmed to be sufficiently small, and the D0–D0 mixing parame-
ter (the SM prediction is xD < 10−3 [11]) must not be measured
by the present experiments, if there is no new physics. By con-
traries, if sizable mixing is measured, new physics should exist
and make main contributions. It is interesting that recently the
D0–D0 has indeed been measured by the BaBar and Belle Col-
laborations [12,13], which may be a signature of existence of
new physics. He and Valencia [14] suggested that the mixing is
due to the FCNC in the up-type-quark sector for non-universal
Z′ model and obtained constraints on the model parameters
by fitting the data. Instead, we propose that the unparticle sce-
nario is the new physics which is responsible for the observable
D0–D0 mixing.
In this study, we will investigate the effects of the unparti-
cle physics on the neutral meson mixing including B0(s)–B
0
(s),
D0–D0 and K0–K0 mixing and constrain the coupling para-
meter of the concerned interactions between the unparticles and
the SM quarks.
2. M0–M0 mixing in unparticle physics
We start with a brief review about the unparticle scenario. It
is assumed that the scale invariant unparticle fields emerge be-
low an energy scale ΛU which is at the order of TeV [3]. The
interactions of the unparticle with the SM particle are described
by a low energy effective theory. For our purpose, the coupling
of unparticle to quarks is given by following the standard strat-
egy to construct effective interactions as
(1)
cS
Λ
dU
U
q¯ ′γμ(1 − γ5)q∂μOU + cV
Λ
dU−1
U
q¯ ′γμ(1 − γ5)qOμU + h.c.,
where OU and OμU denote the scalar and vector unparticle
fields, respectively. The cS and cV are dimensionless coeffi-
cients. We use the same coupling constants for all the flavors
of quarks for simplicity and the V –A type quark currents are
adopted. The above effective interactions may induce FCNCtransitions and provide new physics contribution to the neutral
meson mixing.
In this study, we are only interested in the effects of the un-
particle field which serves as an intermediate agent in the FCNC
transition, thus it only appears as a propagator with momentum
P and scale dimension dU . The propagator for the scalar unpar-
ticle field is given by [4,5]∫
d4x eiP ·x〈0|TOU (x)OU (0)|0〉
(2)= i AdU
2 sin(dUπ)
1
(P 2 + i)2−dU e
−i(dU−2)π ,
where
(3)AdU =
16π5/2
(2π)2dU
(dU + 1/2)
(dU − 1)(2dU ) .
The function sin(dUπ) in the denominator implies that the scale
dimension dU cannot be integral for dU > 1 in order to avoid
singularity. The phase factor e−i(dU−2)π provides a CP conserv-
ing phase which produces peculiar interference effects in high
energy scattering processes [4], Drell–Yan process [5] and CP
violation in B decays [7]. The propagator for the vector unpar-
ticle is similarly given by∫
d4x eiP ·x〈0|TOμU (x)OνU (0)|0〉
(4)= i AdU
2 sin(dUπ)
−gμν + PμP ν/P 2
(P 2 + i)2−dU e
−i(dU−2)π ,
where the transverse condition ∂μOμU = 0 is used.
The neutral meson is denoted by M0(qq¯ ′) and its antiparticle
M0(q ′q¯) where q and q ′ belong to the same up- or down-
type but different flavors. The mixing occurs via a transition
qq¯ ′ → q ′q¯ at the quark level. In the SM, these FCNC processes
can only be realized at loop orders. The lowest contribution
which results in the M0–M0 mixing is the box diagrams. With
the unparticle scenario, the FCNC transitions can occur at tree
level and they are depicted in Fig. 1. The double dashed lines
represent the exchanged unparticle fields. There are two dia-
grams corresponding to t- and s-channel unparticle-exchanges
which contribute to the M0–M0 mixing.
The M0–M0 mixing is usually described by two parame-
ters: the mass difference 	mM and width difference 	M . The
unparticle physics modifies 	mM and thus changes the SM pre-
dictions. For the heavy mesons Bd,Bs,D, the mass difference
	mM is related to the mixing matrix element MM12 by
(5)	mM ≈ 2
∣∣MM12 ∣∣= 1 ∣∣〈M0∣∣Heff(|	F | = 2)∣∣M0〉∣∣,mMFig. 1. The M0–M0 mixing in unparticle physics. The double dashed lines represent the unparticle fields.
382 X.-Q. Li, Z.-T. Wei / Physics Letters B 651 (2007) 380–383where |	F | = 2 represents |	B| = 2 for the B0–B0 mixing
and |	C| = 2 for the D0–D0 mixing. For the D meson system,
the above relation is valid under the assumption of CP conser-
vation. The effective operators which contribute to 	F = 2 are
Q1 = q¯ ′γμ(1 − γ5)qq¯ ′γ μ(1 − γ5)q,
(6)Q2 = q¯ ′(1 − γ5)qq¯ ′(1 − γ5)q.
We only keep the operators at the tree level and more operators
would emerge if QCD corrections are taken into account.
It is noted that the transferred momentum square for t- and
s-channels are approximately equal, i.e., P 2 ≈ m2M for heavy
meson system.
Now we are able to give the expressions for the mass differ-
ence 	mM . The unparticle physics contribution 	mUM is given
as
(7)	mUM =
5
3
f 2MBˆM
mM
AdU
2| sindUπ |
(
mM
ΛU
)2dU
|cS |2,
for the scalar unparticle and
(8)	mUM =
f 2MBˆM
mM
AdU
2| sindUπ |
(
mM
ΛU
)2dU−2
|cV |2,
for the vector unparticle. Note that in the above expression only
the absolute value of the function sindUπ exists. Our results are
the same as in [7] and slightly different from [6] by a constant
factor. In the above derivations, we have used the relations listed
below [15]
〈
M0
∣∣q¯ ′γμ(1 − γ5)qq¯ ′γ μ(1 − γ5)q∣∣M0〉= 83f 2Mm2MBˆM,
(9)〈M0∣∣q¯ ′(1 − γ5)qq¯ ′(1 − γ5)q∣∣M0〉 = −53f 2Mm2MBˆM,
where fM denotes the decay constant and BˆM is a numerical
factor which is related to the non-perturbative QCD and takes
different values in various models, but as known, is of order of
unity.
Some comments are in order:
(1) The mass difference is proportional to a meson mass
dependent factor m2dUM or m
2dU−2
M which comes from the un-
particle propagator 1
(P 2)2−dU . This is a peculiar effect caused
unparticle physics. The propagator for a heavy massive particle
exchange from other new physics does not depend on the low
energy scale mM in general.
(2) The above analysis is applicable to B0–B0, B0s –B0s and
D0–D0 mixing. For the K-system, there are large uncertainties
due to long-distance effects and the approximations which exist
in the theoretical calculations. Thus we will not use the data on
K0–K0 mixing to constrain the unparticle physics parameters.
(3) In this work, following the method commonly adopted in
literature to study new physics effects, we assume that the new
physics beyond the SM which contributes to the mixing is the
unparticle sector. One can write
(10)	mNPM = 	mexpM − 	mSMM ,
where 	mNPM corresponds to the contribution of new physics,
i.e., the unparticle in this study. The SM prediction on 	mBhas already been precise to two-loop order, and the data are
much more accurate than before thanks to the progress in ex-
perimental measurements at BaBar and Belle. Therefore by the
deviation between the SM prediction and measured value, we
can set a constraint on the parameters for the unparticle sce-
nario.
Considering an extreme case, let us loosen the above restric-
tion, namely, we postulate that the mixing B0–B0 is fully due to
the unparticle contribution and see what constraints we would
obtain on the parameters. Later we will show that such con-
straints are looser than that from that obtained from D0–D0
mixing. Therefore, one may not need to take the constraint on
the unparticle parameters from the data of B0–B0 mixing at all.
(4) Because 	mBs
	mBd
= 34  1, B0s –B0s mixing provides a
looser constraint compared to the B0–B0 case.
The unknown parameters about the unparticles are: ΛU ,
dU and cS(cV ). In the numerical results, we fix the value of
ΛU by ΛU = 1 TeV. Other input parameters are: fB
√
BˆB =
0.2 GeV [16], fD
√
BˆD = 0.2 GeV [15], 	mBd = 0.507 ps−1
[17]. The recent experiment carried out by the Belle Collabora-
tions sets xD = 	mDD = (0.80 ± 0.29(stat.) ± 0.17(syst.))% for
the D0–D0 [13]. We use xD < 10−2 as the upper bound.
At first, we consider the case with dU = 3/2 and constrain cS
and cV from B0–B0 and D0–D0 mixing. Table 1 lists the upper
bounds for the coupling parameters cS and cV . The bounds ob-
tained from D0–D0 are more stringent than that from B0–B0
especially for the vector coupling cV . This confirms our expec-
tation in the introduction. The bounds obtained from D0–D0
mixing are: |cS | < 2.1 × 10−2 and |cV | < 5.0 × 10−4.
Then we consider the case with fixed cS , cV and study the
dependence of the D0–D0 mixing parameter xD on the scale
dimension dU . Figs. 2 and 3 plot the dependence within the
Table 1
The upper bounds of |cS | and |cV | with ΛU = 1 TeV and dU = 3/2
From B-system From D-system
|cS | 3.4 × 10−2 2.1 × 10−2
|cV | 2.3 × 10−3 5.0 × 10−4
Fig. 2. The D0–D0 mixing parameter xD versus unparticle scale dimension
(1 < dU < 2). The solid line is given for |cS | = 1 × 10−2 and the dashed line
for |cS | = 2 × 10−2.
X.-Q. Li, Z.-T. Wei / Physics Letters B 651 (2007) 380–383 383Fig. 3. The D0–D0 mixing parameter xD versus unparticle scale dimension
(1 < dU < 2). The solid line is given for |cV | = 2 × 10−5 and the dashed line
for cV = 5 × 10−5.
Fig. 4. The D0–D0 mixing parameter xD versus unparticle scaling dimension
(2 < dU < 3). The solid line is given for |cS | = 10 and the dashed line for
|cS | = 20.
Fig. 5. The D0–D0 mixing parameter xD versus unparticle scaling dimension
2 < dU < 3. The solid line is given for |cV | = 2 × 10−2 and the dashed line for
|cV | = 5 × 10−2.
parameter range 1 < dU < 2. We find that xD is very sensitive
to dU and decreases rapidly to zero as dU increases.
Moreover, we also investigate the case with extending the
scale dimension to the region 2 < dU < 3 and depict the de-
pendence of xD on dU in Figs. 4 and 5. There is no principal
difference compared to the 1 < dU < 2 case except a consider-able change for the coupling parameters cS and cV which are
required to fit the data.
3. Conclusions
We have investigated the new physics effects from scale in-
variant unparticle sectors on the mixing of B0–B0 and D0–D0.
The exchange of unparticle induces the FCNC transitions at tree
level and provides new contribution to the mass difference of
the meson mass eigenstates. In principle, FCNC transitions may
be caused by other new physics effects which contain heavy
massive particles and break the scale invariance. We observe a
peculiar effect caused by the exchange of unparticle: the mix-
ing parameter depends non-trivially on the neutral meson mass.
This dependence might not occur for the heavy massive particle
exchange from other new physics. We use the data on B0–B0
and D0–D0 mixing to constrain the parameters in unparticle
scenario. We find that the D0–D0 mixing provides the most
stringent constraint on the coupling of the scalar and vector un-
particles to the SM quarks. The upper bounds we obtained from
D0–D0 mixing are: |cS | < 2.1 × 10−2 and |cV | < 5.0 × 10−4
if we set the energy scale ΛU = 1 TeV and scale dimension
dU = 3/2. The dependence of scale dimension dU shows that
the mixing parameter is sensitive to the scale dimension and
decreases rapidly by almost two orders of magnitude. The ob-
tained parameters may have important effects on CP violation
in B and D decays.
Acknowledgements
Z. Wei would like to thank Guo-Huai Zhu and Chuan-Hung
Chen for valuable discussions. This work was supported in part
by NNSFC under contract No. 10475042.
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Unparticle physics effects on D0–D0¯ mixing 

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Unparticle Physics Effects on D0-anti-D0 Mixing

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